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The last ESIS blog about how surprisingly few scientists are willing/able to share their experimental data, received an unexpectedly large interest. Directly after the publication another iMechanica blogger took the same theme but he put the focus on results produced at numerical analyses that are presented with insufficient information. While reading, my spontaneous guess was that one obstacle to do right could be the widespread use of commercial non-open codes. The least that then could be done is to demonstrate the ability of the code by comparing results with an exact solution of a simplified example. My fellow blogger also had an interesting reflection regarding differences between theoreticians and computational scientists and it suddenly occurs to me that everything is not black or white. Robert Hooke concealed his results and by writing an anagram, he made sure that he could still take the credit. He didn’t stop at that. When he made his result known he added some ten years to how early he understood the context. And he got away with it.

To some consolation, the EU 8th Framework programme, also called Horizon 2020, finances the OpenAIRE-, and its successor the OpenAIREplus-project that is developed and managed by CERN. The intention is to increase general access to research results with EU support. As a part of this the Zenodo server system was launched. As the observant reader of the previous blog might have seen noted, Zenodo was used by the authors of the survey we discussed in the previous ESIS blog

“Long term availability of raw experimental data in experimental fracture mechanics”, by Patrick Diehl, Ilyass Tabiai, Felix W. Baumann, Daniel Therriault and Martin Levesque, in Engineering Fracture Mechanics, 197 (2018) 21–26, with supplementary materials including all bibtex entries of the papers here  

The purpose of Zenodo is to make sure that there will be enough storage capacity for open access data for everyone. Mandatory for all Horizon2020 financed projects and in first hand all EU financed projects.

I learn from the parallel blog that there are a DataVerse, an openKIM, a Jupyter project and probably much more, in the support of open-access. It seems to me that DataVerse covers the same functionality as Zenodo.  In addition they offer an open-source server with the possibility to set up and run your own server and become integrated in a larger context, which seems very practical. OpenKIM is a systematic collection of atomistic potentials built by users. Jupyter Notebooks yet another open-source project supporting computing in any programming language. They have a written code of conduct. It is not as depressing as it first looks. In essence it summarises your rights and obligations.

It could possibly be better with one single repository or at least one unified system. But why not let a hundred flowers bloom. At the end the solution could be a search engine that covers all or a user’s choice of the open-access repositories. 

Per Ståhle

https://imechanica.org/node/23157

Experimental data availability is a cornerstone for reproducibility in experimental fracture mechanics. This is how the technical note begins, the recently published 

“Long term availability of raw experimental data in experimental fracture mechanics”, by Patrick Diehl, Ilyass Tabiai, Felix W. Baumann, Daniel Therriault and Martin Levesque, in Engineering Fracture Mechanics, 197 (2018) 21–26.

It is five pages that really deserves to be read and discussed. A theory may be interesting but of little value until it has been proven by experiments. All the proof of a theory is in the experiment. What is the point if there is no raw-data for quallity check?

The authors cite another survey that found that 70% of around 1500 researchers failed to reproduce other scientists experiments. As a surprise, the same study find that the common scientists are confident that peer reviewed published experiments are reproducible.

A few years back many research councils around the world demanded open access to all publications emanating from research finansed by them. Open access is fine, but it is much more important to allow examination of the data that is used. Publishers could make a difference by providing space for data from their authors. Those who do not want to disclose their data should be asked for an explanation.

The pragmatic result of the survey is that only 6% will provide data, and you have to ask for it. That is a really disappointing result. The remaining was outdated addresses 22%, no reply 58% and 14% replied but were not willing to share their data. The result would probably still be deeply depressing, but possibly a bit better if I as a researcher only have a single experiment and a few authors to track down. It means more work than an email but on the other hand I don’t have 187 publications that Diehl et al. had. Through friends and former co-authors and some work I think chances are good. The authors present some clever ideas of what could be better than simply email-addresses that are temporary for many researchers.

The authors of the technical note do not know what hindered those 60% who did receive the request and did not reply. What could be the reason for not replying to a message where a colleague asks you about your willingness to share the raw experimental data of a published paper with others? If I present myself to a scientist as a colleague who plan to study his data and instead of studying his behaviour, then chances that he answers increase. I certainly hope that, and at least not the reversed but who knows, life never ceases to surprise. It would be interesting to know what happens. If anyone would like to have a go, I am sure that the author’s of the paper are willing to share the list of papers that they used.

Again, could there be any good reason for not sharing your raw-data with your fellow creatures? What is your opinion? Anyone, the authors perhaps. 

Per Ståhle

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Re: Disclosing raw data

Permalink Submitted by Ajit R. Jadhav on Thu, 2018-08-23 23:19.

Thanks for highlighting the issue.

The idea that raw-data should be available seems quite fine by me, at least on the face of it, though let me hasten to add that personally, I mostly work only in theory, and for that reason, this is more or less a complete non-issue for me. Further, as a programmer, the closest thing that comes to sharing data in my case is: sharing the raw output of programs—though I would have strong objections if all parts of algorithms themselves also were to be disclosed to be able to publish a paper.

As to the latter, I was thinking of this hypothetical scenario. Suppose I invent a new algorithm for speeding up certain simulations. I want to sell that algorithm to some company. I want to get the best possible value for my effort (which is not necessarily the same as the most possible money in the immediate present). But the market is highly fragmented, and so, I don’t want to go through the hassle of contacting every potential customer. So, a good avenue for me is to publish a paper about it. Clearly, here, I can share some data but not all. Especially if the raw data itself can be enough for someone else to figure out at least the kind of algorithm I was using. Data can be a window into the algorithm, which I don’t want to open just yet. How does the proposal work out in this case?

The parallel of the programmer’s case to that of the “hard” experimental research is obvious.

Thus, in some cases, I do anticipate that there could be some IPR-related issues related to the design of the experimental apparatus itself, or of algorithms. Disclosing even just the raw-data could be, in some cases, tantamount to disclosing some other data or ideas that in themselves have some commercial value (present or future), implications for the confidentiality clauses with the clients, and/or patents.

Overall, private organizations pursuing cutting-edge research may have good reasons to pursuing a policy that has both these components: (i) not disclosing the raw data itself, and yet (ii) publishing some of their findings in a summary form, so as to keep the interested public informed about the more distinct stages that their research has reached. The twin policy results, because qua research, it needs to be published (say to gain or retain credibility); qua private data, it anyway cannot a property “owned” by “the public.”

Further, in any case, what is meant by raw-data also needs to be discussed by the research community and clarified. No one would want a worthless explostion in the amount of data. … One sure way to hide “real” information is to cover it under tons of worthless data. You can at least buy some time that way! (To wit: media reports about the Right to Information act in India.)

With all that said, in general, however, I do find the idea that “grant providing organizations should ensure that experimental data by public funded projects is available to the public” very appealing. [Emphasis added]. … Poetic justice! 🙂

Best,

–Ajit

https://imechanica.org/node/22590

It is common practice to obtain stress intensity factors in elastic materials by using Williams series expansions truncated at the r^(-1/2)-stress term. I ask myself, what if both evaluations of experimental and numerical data is improved by including lower order (stronger singularities) terms? The standard truncation is done in a readworthy paper 

“Evaluation of stress intensity factors under multiaxial and compressive conditions using low order displacement or stress field fitting”, R. Andersson, F. Larsson and E. Kabo, in Engineering Fracture Mechanics, 189 (2018) 204–220,

where the authors propose a promising methodology for evaluation of stress intensity factors from asymptotic stress or displacement fields surrounding the crack tip. The focus is on cracks appearing beneath the contact between train wheel and rail and the difficulties that is caused by compression that only allow mode II and III fracture. The proposed methodology is surely applicable to a much larger collection of cases of fracture under high hydrostatic pressure such as at commonplace crushing or on a different length scale at continental transform faults driven by tectonic motion. In the paper they obtain excellent results and I cannot complain about the obtained accuracy. The basis of the analysis is XFEM finit element calculations of which the results are least square fitted to a series of power functions r^n/2. The series is truncated at n=-1 for stresses and 0 for displacements. Lower order terms are excluded. 

We know that the complete series, converges within an annular region between the largest circle that is entirely in the elastic body and the smallest circle that encircles the non-linear region at the crack tip. In the annular ring the complete series is required for convergence with arbitrary accuracy. Outside the annular ring the series diverges and on its boundaries anything can happen. A single term autonomy is established if the stress terms for n<-1 are insignificant on the outer boundary and those for n>-1 are insignificant on the inner boundary. Then only the square root singular term connects the outer boundary to the inner boundary and the crack tip region. Closer to the inner boundary the n≤-1 give the most important contributions and at the outer the n≥-1 are the most important.

I admit that in purely elastic cases the non-linear region at the crack tip is practically a point and all terms n<-1 become insignificant, but here comes my point: Both at evaluation of experiments and numerics the accuracy is often not very good close to the crack tip which often force investigators to exclude data that seem less accurate. This was done in the reviewed paper, where the result from the elements closes to the crack tip was excluded. This is may be the right thing to do but what if n=-2, a r^-1 singularity is included? After all the numerical inaccuracies at the crack tip or the inaccurate measurements or non-linear behaviour at experiments are fading away at larger distances from the crack tip. In the series expansion of stresses in the elastic environment this do appear as finite stress terms for n≤-1.

It would be interesting to hear if there are any thoughts regarding this. The authors of the paper or anyone who wishes express an opinion is encouraged to do so.

Per Ståhle

https://imechanica.org/node/22425

Extreme thermal and electrical conductivity, blocks out almost all gases, stiff as diamond and stronger than anything else. The list of extreme properties seems never ending. The paper

Growth speed of single edge pre-crack in graphene sheet under tension, Jun Hua et al., Engineering Fracture Mechanics 182 (2017) 337–355, 

deals with the fracture mechanical properties of graphene. A sheet of armchair graphene can be stretched up to 15 percent which is much for a crystalline material but not so much when compared with many polymers. The ultimate load, on the other hand, becomes huge almost 100 GPa or more. Under the circumstances, it is problematic to say the least, that the fracture toughness is that of a ceramic, only a few MPam^(1/2). Obviously cracks must be avoided if the high ultimate strength should be useful. Already a few microns deep scratches will bring the strength down to a a few hundred MPa. 

The research group consisting of Jun Hua, Qinlong Liu, Yan Hou, Xiaxia Wu and Yuhui Zhang from the dept. of engineering mechanics, school of science, Xi’an University of Architecture and Technology, Xi’an, China, has studied fast crack growth in a single atomic layer graphene sheet with a pre-crack. They are able to use molecular dynamics simulations to study the kinetics of a quasi-static process. They pair the result with continuum mechanical relations to find crack growth rates. A result that provide confidence is that the fracture toughness obtained from molecular primitives agrees well with what is obtained at experiments. The highlighted results are that the crack growth rate increases with increasing loading rate and decreasing crack length. The tendencies are expected and should be obtained also by continuum mechanical simulations, however then not be first principle and requiring a fracture criterion.

Another major loss would be the possibility to directly observe the details of the fracture process. According to the simulation results the crack runs nicely between two rows of atoms without branching or much disturbances of the ordered lattice. The fracture process itself would not be too exciting if it was not for some occasional minor disorder that is trapped along the crack surfaces. The event does not seem to occur periodically but around one of ten atoms suffers from what the authors call abnormal failure. Remaining at the crack surface are dislocated atoms with increased bond orders. All dislocated atoms are located at the crack surface. The distorted regions surrounding solitary dislocated carbon atoms are small. 

A motivated question would be if the dissipated energy is of the same order of magnitude as the energy required to break the bonds that connects the upper and lower half planes before fracture. Can this be made larger by forcing the crack to grow not along a symmetry plane as in the present study. Without knowing much about the technical possibilities I assume that if two graphene sheets connected to each other rotated so that the symmetry planes do not coincide, the crack would be forced to select a less comfortable path in at least one of the sheets. 

Everyone with comments or questions is cordially invited raise their voice.

Per Ståhle

https://imechanica.org/node/21985

A T-stress is generally not expected to contribute to the stress intensity factor because its contribution to the free energy is the same before and after crack growth. Nothing lost, nothing gained. Some time ago I came across a situation when a T-stress, violates this statement. The scene is the atomic level. As the crack is producing new crack surfaces the elastic stiffness in the few atomic layers closest to the crack plane are modified. This changes the elastic energy which could provide, contribute to or at least modify the energy release rate. If the energy is sufficient depends on the magnitude of the T-stress, the change of the elastic modulus and how many atomic layers that are involved. 

If I should make an estimate it would be that the energy release rate is the change of the T-stress times the fraction of change of the elastic modulus times the square root of the thickness on the affected layer. Assuming that the T-stress is a couple of GPa, the change of the fraction of change of the elastic modulus is 10% and the affected layer is around ten atomic layers one ends up with 100kPa m^(1/2). Fairly small and the stress and its change are taken at its upper limits but still it is there. The only crystalline material I could find is ice with a toughness of the same level. Other materials are affected but require some additional remote load.

Interestingly enough I came across a paper describing a different mechanism leading to a T-stress contribution to the energy release rate. The paper is:

Zi-Cheng Jiang, Guo-Jin Tang, Xian-Fang Li, Effect of initial T-stress on stress intensity factor for a crack in a thin pre-stressed layer, Engineering Fracture Mechanics, pp. 19-27.

This is a really read worthy paper. The reasons for the coupling between the T-stress and the stress intensity factor is made clear by their analysis. The authors have an admirable taste for simple but accurate solutions. The paper describes a crack with a layer of residual stress, that gives a T-stress in the crack tip vicinity. As the crack advances increasing more material end up behind the crack tip rather than in front of it. The elastic energy density caused by the T-stress is larger in front of the tip than it is behind it. The energy released on the way and can only disappear at the singular crack tip, not anywhere else in the elastic material. The reason for the energy release is the assumed buckling in the direction perpendicular to the crack plane. An Euler-Bernoulli beam theory is used to calculate the contribution to the energy release rate.

Having read the paper I realise that in a thin sheet buckling out of its own plane in the presence of a crack and a compressive T-stress there will be energy released that should contribute to crack growth. The buckling will give a more seriously distorted stress state around the crack tip, but never the less. In this case the buckling area would be proportional to the squared crack length in stead of crack length times the height of the layer as in the Jiang et al. paper. The consequence is that the contribution to the stress intensity factor should scale with the T-stress times square root of the crack length.

Suddenly I feel that it would be very interesting to hear if anyone, maybe the authors themselves, know of other mechanisms that could lead to this kind of surprising addition to the energy release rate caused by T-stresses. It would be great if we could add more to the picture. Anyone with information is cordially invited to contribute.

Per Ståhle

https://imechanica.org/node/21796

No doubt the energy release rate comes first. What comes next is proposed in a recently published study that describes a method based on a new constraint parameter Ap. The paper is:

Fracture assessment based on unified constraint parameter for pressurized pipes with circumferential surface cracks, M.Y. Mu, G.Z. Wang, F.Z. Xuan, S.T. Tu, Engineering Fracture Mechanics 175 (2017), 201–218 

The parameter Ap is compared with established parameters like T, Q etc. The application is to pipes with edge cracks. I would guess that it should also apply to other large structures with low crack tip constraint.

As everyone knows, linear fracture mechanics works safely only at small scales of yielding. Despite this, the approach to predict fracture by studying the energy loss at crack growth, using the stress intensity factor KI and its critical limit, the fracture toughness, has been an engineering success story. KI captures the energy release rate at crack growth. This is a well-founded concept that works for technical applications that meet the necessary requirements. The problem is that many or possibly most technical applications hardly do that. The autonomy concept in combination with J-integral calculations, which gives a measure of the potential energy release rate of a stationary crack, widens the range of applications. However, it is an ironin that the J-integral predicts the initiation of crack growth which is an event that is very difficult to observe, while global instability, which is the major concern and surely easy to detect, lacks a basic single parameter theory.

For a working concept, geometry and load case must be classified with a second parameter in addition to KI or J. The most important quantity is no doubt the energy release rate, but what is the second most important. Several successful parameters have been proposed. Most of them describe some type of crack tip constraint, such as the T-stress, Q, the stress triaxiality factor h, etc. A recent suggestion that, as it seems to me, have great potential is a measure of the volume exposed to high effective stress, Ap. It was earlier proposed by the present group GZ Wang and co-authors. Ap is defined as the relative size of the region in which the effective stress exceeds a certain level. As pointed out by the authors, defects in large engineering structures such as pressure pipes and vessels are often subjected to a significantly lower level of crack tip constraint than what is obtained in laboratory test specimens. The load and geometry belong to an autonomy class to speak the language of KB Broberg in his book “Fracture and Cracks”. The lack of a suitable classifying parameter is covered by Ap.

The supporting idea is that KI or J describe the same series of events that lead to fracture both in the lab and in the application if the situations meet the same class requirements, i.e. in this case have the same Ap. The geometry and external loads are of course not the same, while a simpler and usually smaller geometry is the very idea of the lab test. The study goes a step further and proposes a one-parameter criterion that combines the KI or J with Ap by correlation with data.

The method is reinforced by several experiments that show that the method remains conservative, while still avoiding too conservative predictions. The latter of course makes it possible to avoid unnecessary disposal and replacement or repair of components. The authors’ conclusions are based on experience of a particular type of application. I like the use of the parameter. I guess more needs to be done extensively map of the autonomy classes that is covered by the method. I am sure the story does not end here.

A few questions could be sent along: Like “Is it possible to describe or give name to the second most important quantity after the energy release rate?” The paper mentions that statistical size effects and loss of constraint could affect Ap. Would it be possible to do experiments that separates the statistical effect from the loss of constraint? Is it required or even interesting?  

It would be interesting to hear from the authors or anyone else who would like to discuss or comment the paper, the proposed method, the parameter or anything related. 

Per Ståhle

https://imechanica.org/node/21722

Everyone loves an elegant engineering solution. It is particularly true when the alternatives are terrifying. In the paper:

”Brittle crack propagation/arrest behaviour in steel plate – Part I: Model formulation” by Kazuki Shibanuma, Fuminori Yanagimoto, Tetsuya Namegawa, Katsuyuki Suzuki, Shuji Aihara in Engineering Fracture Mechanics, 162 (2016) 324-340.

a team from University of Tokyo proposes a model for prediction of the arrest of propagating brittle cracks in steel plates. The approach, in spite of its simplicity, captures the physics of the fracture process. The model formulates the energy release rate in simple and comprehensible terms and gives accurate predictions. The theory is validated on several experiments described in a subsequent paper, a ”Part II: Experiments and model validation” also in Engineering Fracture Mechanics. The characteristics are those of a pilot study with the goal to provide a design tool for predicting crack arrest in steel plates.

In the model, the energy to complete the fracture process is at most what is left of the released energy when the work of plastic deformation and the part of the kinetic energy that is reflected away from crack tip region have been covered. The energy dissipation at plastic deformation is reduced at increasing crack tip velocity while the opposite applies to the dissipated kinetic energy. The energy required for the fracture processes is supposed to be constant. If it at some velocity is more than required then the energy is in balance only at a single stable higher crack tip velocity. If crack growth is initiated then the crack accelerates until the energy balance is obtained. When the crack subsequently loose driving force or require additional work, caused, e.g., by elevated temperature which decreases the material viscosity or by whatever, the crack decelerates until zero velocity or until the minimum energy release rate is obtained and the crack arrest comes abruptly.

Surface-ligaments are assumed to consume a serious part of the available energy. The slower the crack grows the wider these ligaments become which rapidly increases the plastic dissipation. Finally the energy balance and the stability of the crack tip velocity cannot be maintained and the crack will come to a stop.

Considering that one has to keep track of the complicated sequence of processes that keep the crack growing, it seemed obvious to me that this would end up in a horrible and time consuming analysis. Then, to my surprise, the investigators present an ingenious solution that simplifies the analysis a lot. It is based on three assumptions: 1) that the crack front is assumed to be straight through the plate, 2) that the unbroken side-ligaments are regarded as integrated parts of the crack front, and 3) that the evaluation of the state of the crack front in done at the plate mid-plane.

In the subsequent part II the functionality of the model is verified. The validation is performed on different grades of steel that are exposed to different load levels. The authors believe that this model can be used to establish a design strategy for steel plates. I too believe that, even if more possibly needs to be done to qualify the method as a design standard.

I understand that the authors are familiar with the series of wide plate experiments on crack arrest in very large specimens  (around 11x1x0.1 m3) reported by Naus et al., NUREG/CR-4930 ORNL-6388, Oakridge Laboratories, USA, 1987. 

In the aftermath of the experiments a variety of models where proposed. An interesting observation made by D. Alexander and I.B. Johansson at Oakridge Labs when they examined the crack surfaces was that remains of plastic deformation framed the cloven grains. The guess was that this was remaining parts of broken ligaments between the crack surfaces and that these ligaments were ripped apart during the fracture process. The area covered by these remains was clearly increasing with decreasing crack tip velocity. Just before crack arrest they could cover as much as 10 to 20 % of the ”brittle” part of the crack surface. I have a feeling that this may mean something. The plastic ligaments per se consume large amounts of energy and with increasing fractions they might influence the crack tip velocity at arrest. Only 10% may seem as small or even insignificant, but considering that the plastic ligaments that bridge a crack may consume many times more energy than the pure cleavage of the remaining 90%, even 10% must be important. It would be interesting to know if the authors observed any remains of plastic ligaments. If so, did the fraction of them change in any systematic way during crack growth? 

Any contribution to this blog is gratefully acknowledged.

Per Ståhle

https://imechanica.org/node/20605

A nice demonstration of toughening by introducing multiple secondary cracking of planes parallel with the primary crack is found in the paper:

”Fracture resistance enhancement of layered structures by multiple cracks”  by Stergios Goutianos and Bent F. Sørensen in Engineering Fracture Mechanics, 151 (2016) 92-108.

The 14th paper belong to the category innovative ideas leading to improved composites. We already know of combinations of hard/soft, stiff/weak or brittle/ductile materials that are used to obtain some desired properties. The results are not at all limited to what is set by the pure materials themselves. It has been shown that cracks intersecting soft material layers are exposed to elevated fracture resistances (see eg. the paper 9 blog). Differences in stiffness can be used to improve fatigue and fracture mechanical properties as found in studies by Surresh, Sou, Cominou, He, Hutchinson, and others. Weak interfaces can be used to diverge or split a crack on an intersecting path. A retardation is caused by the additional energy consumed for the extended crack surface area or caused by smaller crack tip driving forces of diverging crack branches. 

A primary crack is confined to grow in a weak layer. The crack tip that is modelled with a cohesive zone remains stationary until the full load carrying capacity of the cohesive forces is reached. Meanwhile the increasing stress across an even weaker adjacent layer also develops a cohesive zone that takes its share of the energy released from the surrounding elastic material. At some point the cohesive capacity is exhausted also here and a secondary crack is initiated. Both cracks are confined to different crack planes and will never coalesce. The continuation may follow different scenarios depending on the distance between the two planes, the relative cohesive properties like cohesive stress, critical crack tip opening, the behaviour at closure etc. of the second layer. All these aspects are studied and discussed in the paper.

The investigators have successfully found a model for how to design the cohesive properties to obtain structures with optimal fracture resistance. Parameters that are manageable in a production process are the ratio of the cohesive properties of the different crack planes and the distance between the them. A theoretical model is formulated. With it they are able to predict whether or not the toughness of a layered structure can be increased by introducing weak layers as described. 

Their results coincide well with the experimental results by Rask and Sørensen (2012) and they have found a model for how to design the cohesive properties to obtain a structure with optimal fracture resistance. Parameters that are manageable in a production process are the ratio of the cohesive properties of the different crack planes and the distance between the them.  

The part that I would like to discuss concerns an estimation of an upper bound of the enhancement of the fracture toughness. The derived theoretical model is based on the J integral taken along a path that ensures path independence. Two different paths are evaluated and compared. Along a remote path the J-value is given as a function of external load and deformation. The structural stiffness is reduced as the crack advances in the direction of the primary crack. In the linear elastic case the J-value is half of the work done by the external load during a unit of crack growth. In an evaluation taken along a local path, J receive contributions from the primary crack tip and the two crack tips of the secondary crack. All three tips are supposed to move a unit of length in the direction of the extending primary crack. 

As observed by the authors the secondary crack does not contribute to the energy release rate while what is dissipated at the propagating foremost crack tip is to the same amount produced at the healing trailing crack tip. Both crack tips propagate in the same direction so that the crack length does not change. 

An observation from the experimental study was that all crack tips have different growth rates and especially the trailing tip of the secondary crack was found to be stationary. Therefore the contribution from that crack tip to the local energy release rate is annulated which leaves less available to the primary crack. To me this seems right. However, when the two remaining advancing crack tips grow does not the respective contributions to J have to be reassessed to reflect their different growth rates? If we assume that the secondary crack grow faster than the primary crack then the enhancing effect is underestimated by the J-integral. Upper bound or lower bound – I can’t decide. I would say that it is a fair estimate of where the fracture resistance will end up. 

In conjunction with the evaluation of the work done by the external load during a ”unit of crack growth” it seems to be an intricate problem to correlate the unit of crack growth with the different crack tip speeds. Some kind of average perhaps.

Any contribution to the blog is gratefully acknowledged.

Per Ståhle

https://imechanica.org/node/20004